Enuerating Equitable Partitions of length up to, 7 By Shalosh B. Ekhad In this article we will give generating functions, as well explicit expressions as sums of quasi-polynomials that enumerate partitions where the sum of the bottom entries exceeds the sum of the top entries, whenever feasible Theorem Number, 1, : Let , f(n), be the number of non-increasing arrays of non-negative integers of length, 3 such that the sum of the, 2, lowest entries is at Least the sum of the top , 1, entries. We have the generating function infinity ----- 3 \ n t ) f(n) t = - --------------------------------------- / 3 2 2 2 ----- (t - 1) (t + 1) (t + t + 1) (t + 1) n = 0 In Maple input notation the right side is: -t^3/(t-1)^3/(t+1)^2/(t^2+t+1)/(t^2+1) f(n), itself can be given explicitly as a sum of the following quasi-polynomials, where a list of polynomials of length k indicates a quasi-polynomial whose value when n is i (mod k) is given by the i-th entry, for i=1..k . 2 n n f(n) = [[1/48 n + 1/16 n - 1/288], [---- + 3/32, - ---- - 3/32], 16 16 [-1/9, -1/9, 2/9], [-1/8, 1/8, 1/8, -1/8]] In Maple input notation the right side is: [[1/48*n^2+1/16*n-1/288], [1/16*n+3/32, -1/16*n-3/32], [-1/9, -1/9, 2/9], [-1/8 , 1/8, 1/8, -1/8]] For the sake of Sloane, the first 40 terms (starting with n=1) are: [0, 0, 1, 0, 1, 1, 2, 1, 3, 2, 4, 3, 5, 4, 7, 5, 8, 7, 10, 8, 12, 10, 14, 12, 16, 14, 19, 16, 21, 19, 24, 21, 27, 24, 30, 27, 33, 30, 37, 33] ------------------------------------------------------- Theorem Number, 2, : Let , f(n), be the number of non-increasing arrays of non-negative integers of length, 4 such that the sum of the, 3, lowest entries is at Least the sum of the top , 1, entries. We have the generating function infinity ----- 3 2 4 \ n (t - t + 1) t ) f(n) t = ---------------------------------------------------- / 4 2 3 2 2 ----- (t - 1) (t + t + 1) (t + 1) (t + 1) (t - t + 1) n = 0 In Maple input notation the right side is: (t^3-t^2+1)*t^4/(t-1)^4/(t^2+t+1)/(t+1)^3/(t^2+1)/(t^2-t+1) f(n), itself can be given explicitly as a sum of the following quasi-polynomials, where a list of polynomials of length k indicates a quasi-polynomial whose value when n is i (mod k) is given by the i-th entry, for i=1..k . 3 2 f(n) = [[1/288 n + 1/64 n + 1/64 n - 1/48], 2 2 [1/192 n - 5/192 n - 1/18, -1/192 n + 5/192 n + 1/18], [1/9, 1/18, -1/6], -3 -1 -1 -1 [1/16, --, --, 3/16], [-1/9, --, 1/18, 1/9, 1/18, --]] 16 16 18 18 In Maple input notation the right side is: [[1/288*n^3+1/64*n^2+1/64*n-1/48], [1/192*n^2-5/192*n-1/18, -1/192*n^2+5/192*n+ 1/18], [1/9, 1/18, -1/6], [1/16, -3/16, -1/16, 3/16], [-1/9, -1/18, 1/18, 1/9, 1/18, -1/18]] For the sake of Sloane, the first 40 terms (starting with n=1) are: [0, 0, 0, 1, 1, 1, 2, 3, 4, 5, 7, 8, 11, 12, 16, 18, 23, 24, 31, 33, 41, 43, 53, 55, 67, 69, 83, 86, 102, 104, 123, 126, 147, 150, 174, 177, 204, 207, 237, 241] ------------------------------------------------------- Theorem Number, 3, : Let , f(n), be the number of non-increasing arrays of non-negative integers of length, 5 such that the sum of the, 3, lowest entries is at Least the sum of the top , 2, entries. We have the generating function infinity ----- \ n 5 / 5 4 2 3 4 3 2 ) f(n) t = - t / ((t - 1) (t + 1) (t + 1) (t + t + t + t + 1) / / ----- n = 0 2 2 2 2 4 2 4 (t + t + 1) (t - t + 1) (t - t + 1) (t + 1)) In Maple input notation the right side is: -t^5/(t-1)^5/(t+1)^4/(t^2+1)^3/(t^4+t^3+t^2+t+1)/(t^2+t+1)^2/(t^2-t+1)^2/(t^4-t ^2+1)/(t^4+1) f(n), itself can be given explicitly as a sum of the following quasi-polynomials, where a list of polynomials of length k indicates a quasi-polynomial whose value when n is i (mod k) is given by the i-th entry, for i=1..k . 4 3 53 2 425 40657 f(n) = [[1/276480 n + 5/27648 n + ----- n + ----- n + -------], [ 18432 27648 2764800 3 25 2 385 3625 1/27648 n + ----- n + ----- n + ------, 18432 27648 110592 3 25 2 385 3625 -1 n n 11 -1/27648 n - ----- n - ----- n - ------], [--, --- + 7/108, - --- - ---], 18432 27648 110592 72 216 216 216 2 517 2 71 [1/1536 n + 5/256 n + ----, 1/1536 n + 5/384 n + ----, 4608 2304 2 517 2 71 -1/1536 n - 5/256 n - ----, -1/1536 n - 5/384 n - ----], 4608 2304 -1 -1 -1 -1 n 25 n n n 25 [--, --, --, --, 4/25], [- --- - ---, - --- - 1/12, --- + 7/216, --- + ---, 25 25 25 25 108 216 216 216 108 216 n n -1 -1 --- + 1/12, - --- - 7/216], [--, 1/16, 0, 0, 1/16, --, 0, 0], 216 216 16 16 -1 -1 -1 -1 -1 -1 [1/36, --, 1/18, --, 1/36, 1/36, --, 1/18, --, 1/36, --, --]] 18 36 36 18 36 36 In Maple input notation the right side is: [[1/276480*n^4+5/27648*n^3+53/18432*n^2+425/27648*n+40657/2764800], [1/27648*n^ 3+25/18432*n^2+385/27648*n+3625/110592, -1/27648*n^3-25/18432*n^2-385/27648*n-\ 3625/110592], [-1/72, 1/216*n+7/108, -1/216*n-11/216], [1/1536*n^2+5/256*n+517/ 4608, 1/1536*n^2+5/384*n+71/2304, -1/1536*n^2-5/256*n-517/4608, -1/1536*n^2-5/ 384*n-71/2304], [-1/25, -1/25, -1/25, -1/25, 4/25], [-1/108*n-25/216, -1/216*n-\ 1/12, 1/216*n+7/216, 1/108*n+25/216, 1/216*n+1/12, -1/216*n-7/216], [-1/16, 1/ 16, 0, 0, 1/16, -1/16, 0, 0], [1/36, -1/18, 1/18, -1/36, 1/36, 1/36, -1/36, 1/ 18, -1/18, 1/36, -1/36, -1/36]] For the sake of Sloane, the first 40 terms (starting with n=1) are: [0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 2, 1, 2, 1, 4, 2, 3, 2, 6, 4, 6, 3, 9, 6, 9, 6, 14, 9, 13, 9, 19, 14, 20, 13, 26, 19, 27, 20] ------------------------------------------------------- Theorem Number, 4, : Let , f(n), be the number of non-increasing arrays of non-negative integers of length, 5 such that the sum of the, 4, lowest entries is at Least the sum of the top , 1, entries. We have the generating function infinity ----- \ n 5 10 9 8 7 6 5 4 3 2 ) f(n) t = - t (t + t + t + t + t + t + t + t + t + t + 1) / ----- n = 0 / 5 4 2 2 2 2 4 / ((t - 1) (t + 1) (t + t + 1) (t - t + 1) (t + 1) (t + 1) / 4 3 2 (t + t + t + t + 1)) In Maple input notation the right side is: -t^5*(t^10+t^9+t^8+t^7+t^6+t^5+t^4+t^3+t^2+t+1)/(t-1)^5/(t+1)^4/(t^2+t+1)/(t^2- t+1)/(t^2+1)^2/(t^4+1)/(t^4+t^3+t^2+t+1) f(n), itself can be given explicitly as a sum of the following quasi-polynomials, where a list of polynomials of length k indicates a quasi-polynomial whose value when n is i (mod k) is given by the i-th entry, for i=1..k . 11 4 11 3 55 2 31559 f(n) = [[----- n + ---- n + ---- n - -------], [ 46080 4608 9216 1382400 3 2 475 1/4608 n + 5/3072 n + 5/384 n + -----, 18432 3 2 475 -1 -1 -1/4608 n - 5/3072 n - 5/384 n - -----], [--, 1/27, --], 18432 54 54 n n 11 n n 11 -1 -1 -1 -1 [- --- + 3/64, --- + ---, --- - 3/64, - --- - ---], [--, --, --, --, 4/25], 128 128 128 128 128 128 25 25 25 25 -1 -1 -1 -1 -1 -1 [1/18, 0, --, --, 0, 1/18], [--, --, --, 1/16, 1/16, 1/16, 1/16, --]] 18 18 16 16 16 16 In Maple input notation the right side is: [[11/46080*n^4+11/4608*n^3+55/9216*n^2-31559/1382400], [1/4608*n^3+5/3072*n^2+5 /384*n+475/18432, -1/4608*n^3-5/3072*n^2-5/384*n-475/18432], [-1/54, 1/27, -1/ 54], [-1/128*n+3/64, 1/128*n+11/128, 1/128*n-3/64, -1/128*n-11/128], [-1/25, -1 /25, -1/25, -1/25, 4/25], [1/18, 0, -1/18, -1/18, 0, 1/18], [-1/16, -1/16, -1/ 16, 1/16, 1/16, 1/16, 1/16, -1/16]] For the sake of Sloane, the first 40 terms (starting with n=1) are: [0, 0, 0, 0, 1, 1, 2, 2, 4, 5, 8, 9, 14, 16, 23, 25, 35, 39, 52, 57, 74, 81, 103, 111, 139, 150, 184, 197, 239, 256, 306, 325, 385, 409, 480, 507, 590, 623, 719, 756] ------------------------------------------------------- Theorem Number, 5, : Let , f(n), be the number of non-increasing arrays of non-negative integers of length, 6 such that the sum of the, 4, lowest entries is at Least the sum of the top , 2, entries. We have the generating function infinity ----- \ n 6 11 10 9 7 3 / 6 ) f(n) t = t (t + t + t + 2 t + t + t + 1) / ((t - 1) / / ----- n = 0 4 3 2 5 2 4 4 2 2 2 (t + t + t + t + 1) (t + 1) (t + 1) (t + 1) (t + t + 1) 2 2 4 2 (t - t + 1) (t - t + 1)) In Maple input notation the right side is: t^6*(t^11+t^10+t^9+2*t^7+t^3+t+1)/(t-1)^6/(t^4+t^3+t^2+t+1)/(t+1)^5/(t^2+1)^4/( t^4+1)^2/(t^2+t+1)^2/(t^2-t+1)^2/(t^4-t^2+1) f(n), itself can be given explicitly as a sum of the following quasi-polynomials, where a list of polynomials of length k indicates a quasi-polynomial whose value when n is i (mod k) is given by the i-th entry, for i=1..k . f(n) = [ 5 19 4 917 3 257 2 11239 4321 [1/1382400 n + ------ n + ------- n + ----- n + ------- n - -------], [ 552960 1658880 73728 1658880 1228800 4 17 3 85 2 343 11617 1/221184 n + ------ n + ----- n - ------ n - ------, 110592 73728 110592 442368 4 17 3 85 2 343 11617 -1/221184 n - ------ n - ----- n + ------ n + ------], 110592 73728 110592 442368 n n 13 5 n [- --- - 1/324, - --- - ---, --- + 1/12], [ 648 162 162 648 3 2 65 353 1/18432 n + 1/1536 n - ---- n - ----, 6144 4608 3 2 51 1/18432 n + 3/2048 n + 5/384 n + ----, 1024 3 2 65 353 -1/18432 n - 1/1536 n + ---- n + ----, 6144 4608 3 2 51 -1 -2 -1/18432 n - 3/2048 n - 5/384 n - ----], [2/25, 1/25, 0, --, --], 1024 25 25 n 11 n 11 n 11 n 11 n [--- + ---, 0, - --- - ---, - --- - ---, 0, --- + ---], [- --- - 5/128, 216 108 216 108 216 108 216 108 128 n n 11 n n n n 11 - --- - 1/8, - --- - ---, --- + 5/64, --- + 5/128, --- + 1/8, --- + ---, 128 128 128 128 128 128 128 128 n -1 -1 -1 -1 - --- - 5/64], [--, 1/18, --, 1/18, 0, 0, 1/36, --, 1/36, --, 0, 0]] 128 36 36 18 18 In Maple input notation the right side is: [[1/1382400*n^5+19/552960*n^4+917/1658880*n^3+257/73728*n^2+11239/1658880*n-\ 4321/1228800], [1/221184*n^4+17/110592*n^3+85/73728*n^2-343/110592*n-11617/ 442368, -1/221184*n^4-17/110592*n^3-85/73728*n^2+343/110592*n+11617/442368], [-\ 1/648*n-1/324, -1/162*n-13/162, 5/648*n+1/12], [1/18432*n^3+1/1536*n^2-65/6144* n-353/4608, 1/18432*n^3+3/2048*n^2+5/384*n+51/1024, -1/18432*n^3-1/1536*n^2+65/ 6144*n+353/4608, -1/18432*n^3-3/2048*n^2-5/384*n-51/1024], [2/25, 1/25, 0, -1/ 25, -2/25], [1/216*n+11/108, 0, -1/216*n-11/108, -1/216*n-11/108, 0, 1/216*n+11 /108], [-1/128*n-5/128, -1/128*n-1/8, -1/128*n-11/128, 1/128*n+5/64, 1/128*n+5/ 128, 1/128*n+1/8, 1/128*n+11/128, -1/128*n-5/64], [-1/36, 1/18, -1/36, 1/18, 0, 0, 1/36, -1/18, 1/36, -1/18, 0, 0]] For the sake of Sloane, the first 40 terms (starting with n=1) are: [0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 2, 2, 4, 4, 6, 4, 9, 9, 13, 10, 20, 18, 26, 21, 37, 33, 47, 39, 65, 59, 81, 67, 107, 95, 129, 112, 169, 151, 201, 173] ------------------------------------------------------- Theorem Number, 6, : Let , f(n), be the number of non-increasing arrays of non-negative integers of length, 6 such that the sum of the, 5, lowest entries is at Least the sum of the top , 1, entries. We have the generating function infinity ----- \ n ) f(n) t = / ----- n = 0 12 11 10 9 8 7 6 5 3 2 6 / (t + t + t + t + t + 2 t + t + t + t + t + t + 1) t / ( / 6 5 2 2 4 4 3 2 2 2 (t - 1) (t + 1) (t + 1) (t + 1) (t + t + t + t + 1) (t + t + 1) 4 3 2 2 (t - t + t - t + 1) (t - t + 1)) In Maple input notation the right side is: (t^12+t^11+t^10+t^9+t^8+2*t^7+t^6+t^5+t^3+t^2+t+1)*t^6/(t-1)^6/(t+1)^5/(t^2+1)^ 2/(t^4+1)/(t^4+t^3+t^2+t+1)/(t^2+t+1)^2/(t^4-t^3+t^2-t+1)/(t^2-t+1) f(n), itself can be given explicitly as a sum of the following quasi-polynomials, where a list of polynomials of length k indicates a quasi-polynomial whose value when n is i (mod k) is given by the i-th entry, for i=1..k . 13 5 37 4 1003 3 31 2 7673 59999 f(n) = [[------- n + ------ n + ------ n + ----- n - ------- n - -------], 1382400 184320 829440 18432 1658880 2764800 4 11 3 17 2 907 43967 [1/184320 n + ----- n - ----- n - ----- n - -------, 92160 18432 61440 2764800 4 11 3 17 2 907 43967 -1/184320 n - ----- n + ----- n + ----- n + -------], 92160 18432 61440 2764800 n 13 n 11 n [- --- - ---, - --- - ---, ---- + 2/27], 162 324 162 324 81 n n n n [--- - 3/256, - --- + 1/128, - --- + 3/256, --- - 1/128], 256 256 256 256 -3 -1 -5 -1 [3/50, 1/25, 1/50, 0, --], [5/108, 1/108, --, ---, ---, 1/27], 25 27 108 108 -1 -1 [1/16, 0, 0, --, --, 0, 0, 1/16], 16 16 -3 -1 -3 -1 -1 [--, --, --, --, 1/25, 3/50, 1/25, 3/50, 1/25, --]] 50 25 50 25 25 In Maple input notation the right side is: [[13/1382400*n^5+37/184320*n^4+1003/829440*n^3+31/18432*n^2-7673/1658880*n-\ 59999/2764800], [1/184320*n^4+11/92160*n^3-17/18432*n^2-907/61440*n-43967/ 2764800, -1/184320*n^4-11/92160*n^3+17/18432*n^2+907/61440*n+43967/2764800], [-\ 1/162*n-13/324, -1/162*n-11/324, 1/81*n+2/27], [1/256*n-3/256, -1/256*n+1/128, -1/256*n+3/256, 1/256*n-1/128], [3/50, 1/25, 1/50, 0, -3/25], [5/108, 1/108, -1 /27, -5/108, -1/108, 1/27], [1/16, 0, 0, -1/16, -1/16, 0, 0, 1/16], [-3/50, -1/ 25, -3/50, -1/25, 1/25, 3/50, 1/25, 3/50, 1/25, -1/25]] For the sake of Sloane, the first 40 terms (starting with n=1) are: [0, 0, 0, 0, 0, 1, 1, 2, 3, 4, 6, 9, 12, 16, 22, 28, 37, 46, 59, 71, 91, 107, 134, 157, 193, 222, 271, 308, 371, 419, 499, 559, 661, 734, 860, 952, 1106, 1216, 1405, 1537] ------------------------------------------------------- Theorem Number, 7, : Let , f(n), be the number of non-increasing arrays of non-negative integers of length, 7 such that the sum of the, 4, lowest entries is at Least the sum of the top , 3, entries. We have the generating function infinity ----- \ n 7 / 7 6 2 4 2 4 ) f(n) t = - t / ((t - 1) (t + 1) (t + t + 1) (t - t + 1) / / ----- n = 0 6 5 4 3 2 2 4 4 3 8 (t + t + t + t + t + t + 1) (t + 1) (t + 1) (t + 1) 4 2 2 8 4 6 3 6 3 (t - t + 1) (t - t + 1) (t + t + 1) (t - t + 1)) In Maple input notation the right side is: -t^7/(t-1)^7/(t+1)^6/(t^2+t+1)^4/(t^2-t+1)^4/(t^6+t^5+t^4+t^3+t^2+t+1)/(t^2+1)^ 4/(t^4+1)^3/(t^8+1)/(t^4-t^2+1)^2/(t^8-t^4+1)/(t^6+t^3+1)/(t^6-t^3+1) f(n), itself can be given explicitly as a sum of the following quasi-polynomials, where a list of polynomials of length k indicates a quasi-polynomial whose value when n is i (mod k) is given by the i-th entry, for i=1..k . 6 11 5 389 4 385 3 f(n) = [[1/20065812480 n + --------- n + --------- n + ------- n 955514880 382205952 8957952 185927 2 3564407 2626371563 5 + --------- n + --------- n + ------------], [1/955514880 n 212336640 477757440 168552824832 77 4 1015 3 168707 2 2762767 25826185 + --------- n + -------- n + --------- n + --------- n + ----------, 382205952 71663616 382205952 477757440 1146617856 5 77 4 1015 3 168707 2 2762767 -1/955514880 n - --------- n - -------- n - --------- n - --------- n 382205952 71663616 382205952 477757440 25826185 3 41 2 481 118007 - ----------], [1/559872 n + ------ n + ----- n + -------, 1146617856 186624 62208 1679616 2 385 54217 -5/186624 n - ------ n - -------, 186624 1679616 3 2 529 31895 -1/559872 n - 1/5184 n - ----- n - ------], [ 93312 839808 3 2 4615 7609 -1/884736 n - 7/49152 n - ------ n - ------, 884736 147456 3 35 2 1499 4795 1/884736 n + ------ n + ------ n + ------, 294912 442368 221184 3 2 4615 7609 1/884736 n + 7/49152 n + ------ n + ------, 884736 147456 3 35 2 1499 4795 -1/884736 n - ------ n - ------ n - ------], [ 294912 442368 221184 3 2 1723 51253 1/559872 n + 5/20736 n + ------ n + ------, 186624 559872 3 77 2 815 54131 1/279936 n + ------ n + ----- n + ------, 186624 62208 559872 3 2 361 1439 1/559872 n + 1/5832 n + ----- n + ------, 93312 279936 3 2 1723 51253 -1/559872 n - 5/20736 n - ------ n - ------, 186624 559872 3 77 2 815 54131 -1/279936 n - ------ n - ----- n - ------, 186624 62208 559872 3 2 361 1439 -1/559872 n - 1/5832 n - ----- n - ------], 93312 279936 -1 -1 -1 -1 -1 -1 2 41 3871 [--, --, --, --, --, --, 6/49], [-1/12288 n - ---- n - -----, 49 49 49 49 49 49 6144 36864 2 679 2 43 4375 -1/12288 n - 3/512 n - ----, -1/12288 n - ---- n - -----, 9216 6144 36864 2 17 287 2 41 3871 1/12288 n + ---- n + ----, 1/12288 n + ---- n + -----, 3072 4608 6144 36864 2 679 2 43 4375 1/12288 n + 3/512 n + ----, 1/12288 n + ---- n + -----, 9216 6144 36864 2 17 287 -1 -1 -1 -1 -1/12288 n - ---- n - ----], [1/27, 0, 1/27, --, 0, --, --, 0, --], [ 3072 4608 54 54 54 54 n 67 n n n 35 n 17 - ---- - ----, ---- + 5/144, - --- - 7/144, --- + ---, - ---- - ----, 1728 1728 1728 864 864 864 1728 1728 n n 67 n n n 35 ---- + 5/864, ---- + ----, - ---- - 5/144, --- + 7/144, - --- - ---, 1728 1728 1728 1728 864 864 864 n 17 n -1 -1 -1 -1 -1 ---- + ----, - ---- - 5/864], [--, --, 1/32, --, 1/32, --, 1/32, --, 1/32, 1728 1728 1728 32 32 32 32 32 -1 -1 -1 -1 -1 -1 1/32, --, 1/32, --, 1/32, --, 1/32], [--, 0, 1/27, --, 0, 1/54, 1/54, 0, --, 32 32 32 27 54 54 -1 -1 -1 -1 -1 -1 1/27, 0, --, 1/54, 0, --, --, 0, 1/54], [1/36, --, --, 1/72, 1/72, --, 1/72, 27 54 54 72 72 36 -1 -1 -1 -1 -1 -1 -1 1/36, --, --, 1/36, 1/72, --, 1/72, 1/72, --, --, 1/36, --, --, 1/72, 1/72, 72 72 36 72 72 72 36 -1 -1 --, --]] 36 72 In Maple input notation the right side is: [[1/20065812480*n^6+11/955514880*n^5+389/382205952*n^4+385/8957952*n^3+185927/ 212336640*n^2+3564407/477757440*n+2626371563/168552824832], [1/955514880*n^5+77 /382205952*n^4+1015/71663616*n^3+168707/382205952*n^2+2762767/477757440*n+ 25826185/1146617856, -1/955514880*n^5-77/382205952*n^4-1015/71663616*n^3-168707 /382205952*n^2-2762767/477757440*n-25826185/1146617856], [1/559872*n^3+41/ 186624*n^2+481/62208*n+118007/1679616, -5/186624*n^2-385/186624*n-54217/1679616 , -1/559872*n^3-1/5184*n^2-529/93312*n-31895/839808], [-1/884736*n^3-7/49152*n^ 2-4615/884736*n-7609/147456, 1/884736*n^3+35/294912*n^2+1499/442368*n+4795/ 221184, 1/884736*n^3+7/49152*n^2+4615/884736*n+7609/147456, -1/884736*n^3-35/ 294912*n^2-1499/442368*n-4795/221184], [1/559872*n^3+5/20736*n^2+1723/186624*n+ 51253/559872, 1/279936*n^3+77/186624*n^2+815/62208*n+54131/559872, 1/559872*n^3 +1/5832*n^2+361/93312*n+1439/279936, -1/559872*n^3-5/20736*n^2-1723/186624*n-\ 51253/559872, -1/279936*n^3-77/186624*n^2-815/62208*n-54131/559872, -1/559872*n ^3-1/5832*n^2-361/93312*n-1439/279936], [-1/49, -1/49, -1/49, -1/49, -1/49, -1/ 49, 6/49], [-1/12288*n^2-41/6144*n-3871/36864, -1/12288*n^2-3/512*n-679/9216, -\ 1/12288*n^2-43/6144*n-4375/36864, 1/12288*n^2+17/3072*n+287/4608, 1/12288*n^2+ 41/6144*n+3871/36864, 1/12288*n^2+3/512*n+679/9216, 1/12288*n^2+43/6144*n+4375/ 36864, -1/12288*n^2-17/3072*n-287/4608], [1/27, 0, 1/27, -1/54, 0, -1/54, -1/54 , 0, -1/54], [-1/1728*n-67/1728, 1/1728*n+5/144, -1/864*n-7/144, 1/864*n+35/864 , -1/1728*n-17/1728, 1/1728*n+5/864, 1/1728*n+67/1728, -1/1728*n-5/144, 1/864*n +7/144, -1/864*n-35/864, 1/1728*n+17/1728, -1/1728*n-5/864], [-1/32, -1/32, 1/ 32, -1/32, 1/32, -1/32, 1/32, -1/32, 1/32, 1/32, -1/32, 1/32, -1/32, 1/32, -1/ 32, 1/32], [-1/27, 0, 1/27, -1/54, 0, 1/54, 1/54, 0, -1/54, 1/27, 0, -1/27, 1/ 54, 0, -1/54, -1/54, 0, 1/54], [1/36, -1/72, -1/72, 1/72, 1/72, -1/36, 1/72, 1/ 36, -1/72, -1/72, 1/36, 1/72, -1/36, 1/72, 1/72, -1/72, -1/72, 1/36, -1/72, -1/ 36, 1/72, 1/72, -1/36, -1/72]] For the sake of Sloane, the first 40 terms (starting with n=1) are: [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 2, 1, 2, 1, 2, 0, 3, 2, 3, 2, 3, 2, 7, 3, 5, 3, 6, 3, 10, 7, 11, 5] ------------------------------------------------------- Theorem Number, 8, : Let , f(n), be the number of non-increasing arrays of non-negative integers of length, 7 such that the sum of the, 5, lowest entries is at Least the sum of the top , 2, entries. We have the generating function infinity ----- \ n 48 32 28 40 30 43 ) f(n) t = - (1 + t + t + 33 t + 41 t + 11 t + 38 t + 4 t / ----- n = 0 45 21 37 5 6 12 8 14 29 + 2 t + 27 t + 10 t + 3 t + 6 t + 19 t + 8 t + 25 t + 25 t 17 20 41 42 2 3 4 47 22 + 22 t + 40 t + 5 t + 8 t + 3 t + 2 t + 4 t + t + 43 t 15 31 23 26 19 27 44 36 + 20 t + 24 t + 29 t + 43 t + 26 t + 28 t + 6 t + 21 t 34 25 11 10 9 7 35 33 + 27 t + 28 t + 12 t + 13 t + 7 t + 5 t + 16 t + 19 t 39 46 38 13 16 24 18 7 / + 8 t + 3 t + 15 t + 15 t + 30 t + 44 t + 35 t ) t / ( / 7 6 2 5 4 3 2 2 2 2 (t - 1) (t + 1) (t + 1) (t + t + t + t + 1) (t + t + 1) 2 2 4 2 4 2 4 3 2 2 (t - t + 1) (t - t + 1) (t + 1) (t - t + t - t + 1) 8 6 4 2 8 6 5 4 3 2 (t - t + t - t + 1) (t + 1) (t + t + t + t + t + t + 1)) In Maple input notation the right side is: -(1+t+t^48+33*t^32+41*t^28+11*t^40+38*t^30+4*t^43+2*t^45+27*t^21+10*t^37+3*t^5+ 6*t^6+19*t^12+8*t^8+25*t^14+25*t^29+22*t^17+40*t^20+5*t^41+8*t^42+3*t^2+2*t^3+4 *t^4+t^47+43*t^22+20*t^15+24*t^31+29*t^23+43*t^26+26*t^19+28*t^27+6*t^44+21*t^ 36+27*t^34+28*t^25+12*t^11+13*t^10+7*t^9+5*t^7+16*t^35+19*t^33+8*t^39+3*t^46+15 *t^38+15*t^13+30*t^16+44*t^24+35*t^18)*t^7/(t-1)^7/(t+1)^6/(t^2+1)^5/(t^4+t^3+t ^2+t+1)^2/(t^2+t+1)^2/(t^2-t+1)^2/(t^4-t^2+1)/(t^4+1)^2/(t^4-t^3+t^2-t+1)^2/(t^ 8-t^6+t^4-t^2+1)/(t^8+1)/(t^6+t^5+t^4+t^3+t^2+t+1) f(n), itself can be given explicitly as a sum of the following quasi-polynomials, where a list of polynomials of length k indicates a quasi-polynomial whose value when n is i (mod k) is given by the i-th entry, for i=1..k . 857 6 2251 5 55801 4 5333 3 f(n) = [[----------- n + --------- n + ---------- n + -------- n 18579456000 884736000 1061683200 10616832 253801 2 9156673 416802641 179 5 + --------- n + ---------- n - ------------], [--------- n 117964800 2654208000 780337152000 884736000 2959 4 6329 3 95257 2 10859777 24451943 + --------- n + -------- n + --------- n + ---------- n + ----------, 353894400 53084160 117964800 2654208000 1769472000 179 5 2959 4 6329 3 95257 2 10859777 - --------- n - --------- n - -------- n - --------- n - ---------- n 884736000 353894400 53084160 117964800 2654208000 24451943 n -5 n 23 - ----------], [--- + 7/486, ----, - --- - ----], [ 1769472000 648 1944 648 1944 11 4 3 1747 2 499 430447 ------- n + 5/98304 n + ------- n + ----- n - --------, 5898240 2949120 98304 88473600 11 4 19 3 61 2 91 216667 ------- n + ------ n + ------ n - ----- n - -------, 5898240 368640 294912 20480 5529600 11 4 3 1747 2 499 430447 - ------- n - 5/98304 n - ------- n - ----- n + --------, 5898240 2949120 98304 88473600 11 4 19 3 61 2 91 216667 - ------- n - ------ n - ------ n + ----- n + -------], 5898240 368640 294912 20480 5529600 3 n 83 41 3 n 3 n 57 3 n 23 [- ---- - ----, ----, ---- + 1/50, --- + ---, - --- - ---], 1000 1000 1000 1000 500 500 500 250 n n 65 n 11 n n 65 n 11 [--- + 4/81, --- + ---, --- + ---, - --- - 4/81, - --- - ---, - --- - ---], 648 324 648 648 216 648 324 648 648 216 -1 -1 -1 -1 -1 -1 3 n 51 n -1 n 19 [--, --, --, --, --, --, 6/49], [--- + ---, --- + 5/128, ---, - --- - ---, 49 49 49 49 49 49 512 512 512 256 128 256 3 n 51 n n 19 9 n 93 3 n 63 - --- - ---, - --- - 5/128, 1/256, --- + ---], [- ---- - ----, - --- - ----, 512 512 512 128 256 1000 1000 500 1000 9 n 12 3 n 3 n 9 n 93 3 n 63 - ---- - ---, - --- - 7/100, --- + 7/125, ---- + ----, --- + ----, 1000 125 500 500 1000 1000 500 1000 9 n 12 3 n 3 n ---- + ---, --- + 7/100, - --- - 7/125], 1000 125 500 500 -1 -1 -1 -1 -1 -1 [---, ---, --, --, ---, ---, 1/108, 1/108, 1/54, 1/54, 1/108, 1/108], 108 108 54 54 108 108 -1 -1 -3 [0, 0, 0, 1/16, 0, 0, 1/16, 0, 0, 0, 0, --, 0, 0, --, 0], [1/50, ---, 3/100, 16 16 100 -1 -3 -1 -1 -3 -1 -3 --, 1/50, ---, 3/100, --, 1/50, 1/50, --, 3/100, ---, 1/50, --, 3/100, ---, 50 100 50 50 100 50 100 -1 -1 1/50, --, --]] 50 50 In Maple input notation the right side is: [[857/18579456000*n^6+2251/884736000*n^5+55801/1061683200*n^4+5333/10616832*n^3 +253801/117964800*n^2+9156673/2654208000*n-416802641/780337152000], [179/ 884736000*n^5+2959/353894400*n^4+6329/53084160*n^3+95257/117964800*n^2+10859777 /2654208000*n+24451943/1769472000, -179/884736000*n^5-2959/353894400*n^4-6329/ 53084160*n^3-95257/117964800*n^2-10859777/2654208000*n-24451943/1769472000], [1 /648*n+7/486, -5/1944, -1/648*n-23/1944], [11/5898240*n^4+5/98304*n^3+1747/ 2949120*n^2+499/98304*n-430447/88473600, 11/5898240*n^4+19/368640*n^3+61/294912 *n^2-91/20480*n-216667/5529600, -11/5898240*n^4-5/98304*n^3-1747/2949120*n^2-\ 499/98304*n+430447/88473600, -11/5898240*n^4-19/368640*n^3-61/294912*n^2+91/ 20480*n+216667/5529600], [-3/1000*n-83/1000, 41/1000, 3/1000*n+1/50, 3/500*n+57 /500, -3/500*n-23/250], [1/648*n+4/81, 1/324*n+65/648, 1/648*n+11/216, -1/648*n -4/81, -1/324*n-65/648, -1/648*n-11/216], [-1/49, -1/49, -1/49, -1/49, -1/49, -\ 1/49, 6/49], [3/512*n+51/512, 1/512*n+5/128, -1/256, -1/128*n-19/256, -3/512*n-\ 51/512, -1/512*n-5/128, 1/256, 1/128*n+19/256], [-9/1000*n-93/1000, -3/500*n-63 /1000, -9/1000*n-12/125, -3/500*n-7/100, 3/500*n+7/125, 9/1000*n+93/1000, 3/500 *n+63/1000, 9/1000*n+12/125, 3/500*n+7/100, -3/500*n-7/125], [-1/108, -1/108, -\ 1/54, -1/54, -1/108, -1/108, 1/108, 1/108, 1/54, 1/54, 1/108, 1/108], [0, 0, 0, 1/16, 0, 0, 1/16, 0, 0, 0, 0, -1/16, 0, 0, -1/16, 0], [1/50, -3/100, 3/100, -1/ 50, 1/50, -3/100, 3/100, -1/50, 1/50, 1/50, -1/50, 3/100, -3/100, 1/50, -1/50, 3/100, -3/100, 1/50, -1/50, -1/50]] For the sake of Sloane, the first 40 terms (starting with n=1) are: [0, 0, 0, 0, 0, 0, 1, 1, 2, 1, 2, 2, 5, 5, 8, 8, 15, 14, 22, 20, 35, 33, 49, 47, 76, 72, 102, 97, 148, 140, 193, 185, 270, 258, 345, 330, 466, 447, 585, 560 ] ------------------------------------------------------- Theorem Number, 9, : Let , f(n), be the number of non-increasing arrays of non-negative integers of length, 7 such that the sum of the, 6, lowest entries is at Least the sum of the top , 1, entries. We have the generating function infinity ----- \ n 7 21 20 19 18 17 16 15 ) f(n) t = - t (t + t + t + 2 t + 2 t + 3 t + 4 t / ----- n = 0 14 13 12 11 10 9 8 7 6 5 + 4 t + 4 t + 5 t + 4 t + 4 t + 4 t + 3 t + 3 t + 3 t + 2 t 4 3 2 / 7 6 2 3 4 + 2 t + 2 t + t + t + 1) / ((t - 1) (t + 1) (t + 1) (t + 1) / 4 3 2 2 2 2 2 4 2 (t + t + t + t + 1) (t + t + 1) (t - t + 1) (t - t + 1) 6 5 4 3 2 4 3 2 (t + t + t + t + t + t + 1) (t - t + t - t + 1)) In Maple input notation the right side is: -t^7*(t^21+t^20+t^19+2*t^18+2*t^17+3*t^16+4*t^15+4*t^14+4*t^13+5*t^12+4*t^11+4* t^10+4*t^9+3*t^8+3*t^7+3*t^6+2*t^5+2*t^4+2*t^3+t^2+t+1)/(t-1)^7/(t+1)^6/(t^2+1) ^3/(t^4+1)/(t^4+t^3+t^2+t+1)/(t^2+t+1)^2/(t^2-t+1)^2/(t^4-t^2+1)/(t^6+t^5+t^4+t ^3+t^2+t+1)/(t^4-t^3+t^2-t+1) f(n), itself can be given explicitly as a sum of the following quasi-polynomials, where a list of polynomials of length k indicates a quasi-polynomial whose value when n is i (mod k) is given by the i-th entry, for i=1..k . 19 6 109 5 563 4 1811 3 239 2 f(n) = [[-------- n + -------- n + ------- n + ------- n - ------- n 77414400 11059200 4423680 3317760 7372800 38287 64314829 - ------- n - ----------], [ 6635520 3251404800 5 19 4 179 3 941 2 44561 29081 1/11059200 n + ------- n + ------- n + ------- n + ------- n + -------, 4423680 3317760 1474560 6635520 2457600 5 19 4 179 3 941 2 44561 29081 -1/11059200 n - ------- n - ------- n - ------- n - ------- n - ------- 4423680 3317760 1474560 6635520 2457600 n 11 n 5 n 35 2 655 ], [--- + ---, --- + 1/324, - --- - ---], [-1/6144 n - 3/1024 n - -----, 162 216 648 648 648 18432 2 17 2 655 1/6144 n + 5/1536 n + ----, 1/6144 n + 3/1024 n + -----, 2304 18432 2 17 -1 -1 n -1/6144 n - 5/1536 n - ----], [--, 1/50, 0, 1/50, --], [- --- - 1/648, 2304 50 50 324 n 13 n n n 13 n - --- + ---, --- + 1/24, --- + 1/648, --- - ---, - --- - 1/24], 648 324 648 324 648 324 648 -1 -1 -1 -1 -1 -1 -1 -1 [--, --, --, --, --, --, 6/49], [0, 1/32, 1/32, 0, 0, --, --, 0], 49 49 49 49 49 49 32 32 -1 -3 -3 -1 [3/50, 1/50, 0, --, --, --, --, 0, 1/50, 3/50], 50 50 50 50 -1 -1 -1 -1 -1 -1 [--, --, --, --, --, 1/36, 1/36, 1/18, 1/18, 1/36, 1/36, --]] 36 18 18 36 36 36 In Maple input notation the right side is: [[19/77414400*n^6+109/11059200*n^5+563/4423680*n^4+1811/3317760*n^3-239/7372800 *n^2-38287/6635520*n-64314829/3251404800], [1/11059200*n^5+19/4423680*n^4+179/ 3317760*n^3+941/1474560*n^2+44561/6635520*n+29081/2457600, -1/11059200*n^5-19/ 4423680*n^4-179/3317760*n^3-941/1474560*n^2-44561/6635520*n-29081/2457600], [1/ 162*n+11/216, 1/648*n+1/324, -5/648*n-35/648], [-1/6144*n^2-3/1024*n-655/18432, 1/6144*n^2+5/1536*n+17/2304, 1/6144*n^2+3/1024*n+655/18432, -1/6144*n^2-5/1536* n-17/2304], [-1/50, 1/50, 0, 1/50, -1/50], [-1/324*n-1/648, -1/648*n+13/324, 1/ 648*n+1/24, 1/324*n+1/648, 1/648*n-13/324, -1/648*n-1/24], [-1/49, -1/49, -1/49 , -1/49, -1/49, -1/49, 6/49], [0, 1/32, 1/32, 0, 0, -1/32, -1/32, 0], [3/50, 1/ 50, 0, -1/50, -3/50, -3/50, -1/50, 0, 1/50, 3/50], [-1/36, -1/18, -1/18, -1/36, -1/36, 1/36, 1/36, 1/18, 1/18, 1/36, 1/36, -1/36]] For the sake of Sloane, the first 40 terms (starting with n=1) are: [0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 6, 10, 13, 19, 24, 34, 42, 58, 70, 93, 112, 145, 171, 218, 256, 320, 372, 458, 528, 643, 735, 884, 1006, 1198, 1352, 1597, 1795, 2102, 2350] ------------------------------------------------------- It took, 6182.662, seconds to generate this article.